Is it possible to automatically move other strips while dragging another strip?
Well, it's possible. You can use the "Expand to fit" option which is enabled by hitting G for grab and then you can hit G again to toggle "Expand to fit" permanently or hold down ALT to use toggle it temporarily.

When you have three strips with the new one selected but not yet in your final row:



Just hit G to grab and then G again and place it "over" the existing strips:



Confirm with left mouse button (LMB) or Enter and there you have the strip inserted in between the two existing ones:



And to answer your second question: Just snap the handle next to the other strips one with dragging it and holding down Ctrl. Then just Shift-select the other handle and slide both of them with G.

Moving a single handle works with expand to fit as well (even though, the next strip is then moved and not trimmed in length).

Finally you can use Alt+Left Arrow/Right Arrow to swap two adjacent strips.